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Properties

Label 2-65-13.7-c2-0-3
Degree $2$
Conductor $65$
Sign $0.203 - 0.978i$
Analytic cond. $1.77112$
Root an. cond. $1.33083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.27 − 0.340i)2-s + (−1.95 + 3.38i)3-s + (−1.96 + 1.13i)4-s + (1.58 + 1.58i)5-s + (−1.33 + 4.97i)6-s + (7.68 + 2.05i)7-s + (−5.83 + 5.83i)8-s + (−3.15 − 5.47i)9-s + (2.54 + 1.47i)10-s + (−0.797 − 2.97i)11-s − 8.87i·12-s + (−0.398 − 12.9i)13-s + 10.4·14-s + (−8.45 + 2.26i)15-s + (−0.894 + 1.54i)16-s + (20.4 − 11.7i)17-s + ⋯
L(s)  = 1  + (0.635 − 0.170i)2-s + (−0.652 + 1.12i)3-s + (−0.490 + 0.283i)4-s + (0.316 + 0.316i)5-s + (−0.222 + 0.829i)6-s + (1.09 + 0.294i)7-s + (−0.729 + 0.729i)8-s + (−0.351 − 0.608i)9-s + (0.254 + 0.147i)10-s + (−0.0724 − 0.270i)11-s − 0.739i·12-s + (−0.0306 − 0.999i)13-s + 0.747·14-s + (−0.563 + 0.151i)15-s + (−0.0559 + 0.0968i)16-s + (1.20 − 0.693i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.203 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.203 - 0.978i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ 0.203 - 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01140 + 0.822466i\)
\(L(\frac12)\) \(\approx\) \(1.01140 + 0.822466i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.58 - 1.58i)T \)
13 \( 1 + (0.398 + 12.9i)T \)
good2 \( 1 + (-1.27 + 0.340i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (1.95 - 3.38i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (-7.68 - 2.05i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (0.797 + 2.97i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-20.4 + 11.7i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (1.03 - 3.84i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-19.0 - 10.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (9.86 - 17.0i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-21.3 - 21.3i)T + 961iT^{2} \)
37 \( 1 + (13.2 + 49.3i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (52.5 - 14.0i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-61.1 + 35.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (37.3 - 37.3i)T - 2.20e3iT^{2} \)
53 \( 1 + 19.8T + 2.80e3T^{2} \)
59 \( 1 + (-5.90 - 1.58i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (48.4 + 83.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-102. + 27.5i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-11.4 + 42.7i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-16.3 + 16.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 81.0T + 6.24e3T^{2} \)
83 \( 1 + (90.7 + 90.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-32.4 - 121. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (0.0217 - 0.0811i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.75266355209840292890988988084, −13.98803045105097215959551198575, −12.58255299972682518814964731044, −11.46739486056941378377954651815, −10.56503411955230719393198114549, −9.314671982426239593036412813823, −7.917159606191174658337703255370, −5.49957603881061174486852406663, −4.98659527730731066983462299973, −3.35009114041399338362361567948, 1.32976137983315005508893180308, 4.48449091992615548870427330881, 5.67980592149185197790572224563, 6.86692823337664218453268508202, 8.300304897680712048477045705419, 9.857084288564013621299921467195, 11.45036812395469050220953512977, 12.39074728873231474914196967477, 13.30860249245356479502809134032, 14.17586638637082918993109380834

Graph of the $Z$-function along the critical line